Following [1]} we fix a deterministic algorithm
computing a “good” choice of orientations of the edges induced by the
fraternity property. The precise definition of the algorithm is not important
for us, it only matters here that the algorithm runs in time linear in the
size of the input graph (see Lemma REF  below).
With this algorithm fixed, we can now speak of the
1-transitive fraternal augmentation of \(\vec{\textbf {\textbf {G} }}\) .
